

A054434


Number of possible positions in an n X n X n Rubik's cube reachable from the starting position.


14




OFFSET

1,2


COMMENTS

The sequence counts possible positions of the Rubik's cube considering the positions which are related through rotations of the cube as a whole (there are 24 of those) as distinct. At odd n, the orientation of the cube as a whole is usually considered fixed by the central squares of each face (i. e., the cube as a whole cannot be rotated) so there is a difference compared to A075152 only in the case of even n.  Andrey Zabolotskiy, Jun 07 2016


LINKS

Table of n, a(n) for n=1..5.
Francocube forum, [4x4x4] Les maths du 4x4x4
Georges Helm, Rubik's Cube
M. E. Larsen, Rubik's Revenge: The Group Theoretical Solution, Amer. Math. Monthly 92, 381 (1985), DOI:10.2307/2322445.
Christopher Mowla, Math 3900
Robert Munafo, Rubik's Cube and other Cuboid Puzzles
Philippe Picart, Le Rubik's cube
E. Rubik, Rubik Cube Site
Jaap Scherphuis, Puzzle Pages
Xavier Servantie, All about Rubik's cube
Author?, Rubik's Cube
Index entries for sequences related to Rubik cube


FORMULA

From Andrey Zabolotskiy, Jun 24 2016: (Start)
a(n) = A075152(n)*24 if n is even,
a(n) = A075152(n) if n is odd.
a(2) = Sum(A080629) = Sum(A080630). (End)
a(1)=1; a(2)=24*7!*3^6; a(3)=8!*3^7*12!*2^10; a(n)=a(n2)*24^6*(24!/24^6)^(n2).  Herbert Kociemba, Dec 08 2016


EXAMPLE

From Andrey Zabolotskiy, Jun 24 2016 [following Munafo]: (Start)
a(4) = 8! * 3^7 * 24! * 24! / 4!^6 is constituted by:
8! permutation of corners
× (12*2)! permutation of edges
× (6*4)! permutation of centers
× 1 (combination of permutations must be even, but we can achieve what appears to be an odd permutation of the other pieces in the cube by "hiding" a transposition within the indistinguishable pieces of one color)
× 3^8 orientations of corners
/ 3 total orientation of corners must be zero
× 1 (orientations of edges and centers are determined by their position)
/ 4!^6 the four center pieces of each color are indistinguishable
(End)


MATHEMATICA

f[1]=1; f[2]=24*7!3^6; f[3]=8!3^7 12!2^10; f[n_]:=f[n2]*24^6*(24!/24^6)^(n2); Table[f[n], {n, 1, 10}] (* Herbert Kociemba, Dec 08 2016 *)


CROSSREFS

See A075152, A007458 for other versions.
Sequence in context: A003825 A114259 A234981 * A164850 A253269 A227654
Adjacent sequences: A054431 A054432 A054433 * A054435 A054436 A054437


KEYWORD

nonn,nice


AUTHOR

Antreas P. Hatzipolakis


EXTENSIONS

a(4) and a(5) corrected and definition clarified by Andrey Zabolotskiy, Jun 24 2016


STATUS

approved



